We have then, the following rule RULE.-1. Find the whole number which is nearest the root of the number of hundreds. 2. Place this root at the right hand, and subtract its second power from the hundreds, and bring down the other two figures. 3. Double the root of the hundreds, and divide the remainder by it, omitting the right hand digit. 4. Place the result at the right of the root already found; and also at the right of the divisor; multiply the divisor thus increased by the quotient; and if the quotient be the correct figure in the root, the product will be equal to the remainder. Find the second roots of the following numbers: (1) 2601 4096 4356 (10) 5776 4489 (11) 6889 9 5329 (12) 7921 It is required to find the root of 54756. It is easy to see that the root of this number is over 100, for the second power of 100 is only 10,000. It is easy to see, moreover, that the root is between 200 and 300. 2 then, is the number of hundreds in the root. Assume x = the tens in the root, Then 200+x is equal to the root, except the units. (A) 200+x 200+x 40,000+2.200 x + x2◄54756 signifies that the quantity on the left is less than the one on the right. And as 200+x is less than the root of 54756, so 40,000 + 2 200x+x2, the second power of 200+x, is less than 54756. Subtract 40,000 from each side of ◄ in expression (A,) and we have 2.200 x + x2 ◄14756, or (2.200 + x) x◄14756. For if the same number be subtracted from unequals, the remainders are unequal. Assume for the value of x, the largest number of tens which will make 2.200 x + x2 less than 14756. This can be done by a few trials; and the number of experiments may be abridged by observing that 2. 200 x, or 400 x are less than 14756; consequently, 14756 divided by 400 will give a quotient which cannot be less than x. This division gives 3 tens; x then, cannot be more than 3 tens or 30. Substitute 30 for x in the expression above, and we have (400+30) 30, or 430 × 30, or 12900, a number less than 14756: 3 then, is the number of tens in the root. Assume Then And Or Or (230)2+2.230 y + y2 = 54756 (200+30)2+2.230 y + y2= 54756 Or 40,000+2.200 × 30 + 900 +2.230 y + y2 = 54756 40,000+ 12900 +2.230 y + y2 = 54756 12900+2.230 y + y2 = 14756 2.230 y + y2 (2.230+y) y (460 + y) y 1856 = 1856 We may here find y by trial; or since 460 y 1856; 1856 divided by 460 will give a quotient greater than y. Generally this division gives y and a fraction. This division gives 4 and.... a fraction. In order to ascertain whether 4 is the right number, add 4 to 460, and multiply the sum by 4, which gives 1856; 4, then, is the correct value of y. Hence 200+30 +4, or 234 is the root required. We shall exhibit the preceding process, omitting x and y. 54756 (200+30 +4 400+30) 14756 4604) 1856 If there be any remainder, the root cannot be found in whole numbers. And the whole number found, is only the whole number which is nearest the root. Find by this process the roots of the following numbers: (1) 73984 (3) 78961 (2) 76176 (4) 82369 5 (6) 134689 147456 152100 The rule given on page 79 may be extended so as to be applied to finding the root of any number. Find by the rule, the root of the first three figures on the left, if there be an odd number of figures, or the first four if there be an even number. To the remainder bring down the next two figures. Double the whole root already found, and find the next figure as the second was found, by rule. Then bring down the next two figures, and proceed as before. Find the roots of the following numbers: 952576 (3) 904401 826281 SECTION XXXVIII. Extraction of the Second Root, We find the second power of a fraction by taking the second power of its numerator, and the second power of its denominator. Hence we obtain the root of a fraction by extracting the root of its numerator, and the root of its denominator. The root of is 3, the root of is 7, &c. 1. What are the roots of the following fractions? 1 81 6 4 33 144 625 ; ; ; 10; 10; 1000500. Find the second root of . 40 When we cannot find exactly, either the root of the denominator or the numerator, we must change the form of the frac tion, so that the root of the denominator can be found; thus, 4 and = 1. Hence the root of is the same as the root of; which is nearly. That which is near the root of a number is called the approximate root of the number. is called the approximate root of, or 3. 6 6 is the approximate root of, or fo We may obtain a fraction still nearer the root. 60 = 100 10000. 6000 10000 Thus The larger the denominator, the nearer the root found, approximates the true one. And since we may find the root of 100, 10,000, 1,000,000, or of any number expressed by 1 and an even number of cyphers, we may assume the denominator sufficiently large to approximate the root to any required degree of exactness. 3 3. What are the roots of,, and ? By the same process we may approximate the roots of whole numbers. Suppose it is required to find the root of 2. 2=200 200 2000 100 1000 The root of 20000 is 141 or 1. 41 4. Find the approximate roots of 3, of 5, 8, 13, 27. If we wish to obtain the root of a mixed number, which consists of a whole number and a fraction, we may reduce both the whole number and fraction to hundredths, or ten thousandths, &c. 5. It is required to find the approximate root of 23. 23=20%=14%. The root of 248 is 15, or 1. 6. Find the approximate roots of 25, 2, 3, 41, 63, 2413. Substitute these values of x2 and 4 x, in equation (1,) we Substitute these values of 22 and 8x, in equation (1,) and we have y2+8y+16-8y-32-20 = y2-16 20 = y 6 x=y+4= 10 Assume x equal to y united to half the coefficient of x, with the contrary sign, and substitute as in the preceding examples, and we shall have an equation containing only the second power of y; and from which y may be found. In this way we avoid the necessity of completing the square. Find the value of x in the following equations: x2 + 12 x = 28 8. A man sold a horse for 75 dollars, and thereby gained as much per cent. as the horse cost him. How much did the horse cost him? |